Math test

(There are eight questions in this question, full mark 120, examination time 120 minutes).

1. Fill in the blanks (this big question * * 10 small question, 2 points for each small question, 20 points for * *). Please fill in the answer directly on the horizontal line in the question.

1. If -m=4, then m=.

2. The temperature of cold storage A is -5℃ and that of cold storage B is-15℃, so the temperature of cold storage is the highest.

3. The solution set of inequality 3x-9≤0 is.

4. Given that the radii of ⊙O 1 and ⊙O2 are 2 and 4, respectively, and 0 1O2 = 6, the positional relationship between ⊙O 1 and ⊙O2 is.

5. Overlap the right-angle vertices of two right-angle triangular rulers to the position as shown in figure 1. If ∠ AOD = 1 1o, ∠ BOC =

6. When solving equation (x2-5)2-x2+3=0, make x2-5 = y, and the original equation becomes.

7. Fold this figure into a cube. If the values of opposite faces are equal, a set of x and y values are.

8. (Choose an option to answer this question. )

Scheme 1: After pressing the key in turn on the started scientific calculator, the displayed result (the result retains three significant figures) is as follows

Attached key: Option 2: If the volume of the cube is 2004, the side length of the cube (the result retains three significant figures) is.

Attached cubic table

Number 1 2 3 4 5 6 7 8 9

0.20

2.O

20 0.5848

1.260

2.7 14 0.5858

1.262

2.7 19 0.5867

1.264

2.723 0.5877

1.266

2.728 0.5887

1.268

2.732 0.5896

1.270

2.737 0.5 906

1.272

2.74 1 0.59 15

1.274

2.746 0.5925

1.277

2.750 0.5934

1.279

2.75 5

9. Observe the arrangement of the following balls (where ● is a solid ball and ○ is a hollow ball):

●○○●●○○○○○●○○●●○○○○○●○○●●○○○○○●……

From 1 ball to 2004 ball, * * * has solid balls.

10. A telecom company has introduced two ways to charge for mobile phones: one is to rent 20 yuan on a monthly basis, and the other is to rent 0 yuan on a monthly basis. As shown in Figure 3, the functional relationship between one month's outage time t (minutes) and outage cost s (yuan), when the outage time is 150 minutes, the difference between the two methods is yuan.

Second, multiple-choice questions (this big question ***8 small questions, each small question 3 points, ***24 points) Of the four options given in each small question, only one meets the meaning of the question. Please put the serial number of the correct answer in brackets after the question.

1 1. The following operation is correct ().

A.6a+2a=8a2 B. a2÷a2=0

C.a-(a-3)=-3D。 D.a- 1 a2=a

12. Given the line segment AB, take a point C on the extension line of BA to make CA=3AB, then the ratio of line segment CA to line segment CB is ().

A.3:4 B.2:3 C.3:5 D. 1:2

13. Factorizing 4-4a+A2, the correct one is ().

a . 4( 1-a)+a2 b .(2-a)2c .(2-a)(2-a)d .(2+a)2

14. The following proposition is wrong ().

A an equilateral triangle has equal sides and angles. An equilateral triangle is an axisymmetric figure.

C. an equilateral triangle is a figure d with central symmetry. An equilateral triangle has an inscribed circle and an circumscribed circle.

15. As shown in Figure 4, P 1, P2 and P3 are three points on the hyperbola. When these three points are perpendicular to the Y axis, three triangles P 1A 10, P2A20 and P3A30 are obtained. Let their regions be S 1, S2 and S3 respectively.

A.s 1 & lt； S2<S3·S2 & lt； s 1 & lt； S3 C . s 1 & lt； S3<S2 D.S 1=S2=S3

16. The list price of hotel rooms affects the percentage of accommodation. The following table is the average statistical data of a hotel in recent years:

Room rate (RMB)160140120100

The accommodation percentage is 63.8% 74.3% 84. 1% 95%.

In the tourism week, to maximize the income of hotel rooms, the room price should be selected ().

A 160 yuan B 140 yuan C 120 yuan D 100 Yuan.

17. As shown in the figure, ⊙O 1 and ⊙O2 intersect at point A and point B, and the straight line CD passing through point A intersects with ⊙O 1 and ⊙O2 at point C and point D respectively, and the straight line EF passing through point B intersects with ⊙ o657; respectively.

①CE∨DF； ②∠D =∠F； ③ ef = 20 1o2。 There must be ().

A.0 B. 1 C.2 D.3

18. As shown in the figure, if the diameter ⊙0 is AB=8, P is any point on the upper semicircle (except A and B), the bisector of ∠APB intersects ⊙O at C, and the chord EF passes through the middle points M and N of AC and BC, then the length of EF is ().

a4 b . 2 c . 6d . 2

Third, the title of this big question, out of ***76 points, the answer should be written. A proof process or calculus step.

Iii. This topic is entitled ***3 small questions, and the full score is *** 15.

19. (Full score for this small question)

20. (The full score for this short question is 5)

Two fractions are known: A=, B=, where x ≠ 2.

There are three conclusions as follows: ① A = B; ②A and B are reciprocal; ③A and B are reciprocal.

Excuse me, which one is correct? Why?

2 1. (Full score for this small question)

The results of five math tests of students A and B are as follows:

Test (times) 1 2 3 4 5 average variance

A (score) 75 90 96 83 8 1

B (points) 86 70 90 95 84

Please fill in the appropriate figures in the blank of the form, analyze the scores of two students with your statistical knowledge, and write a reasonable suggestion.

Four, this big topic ***2 small questions, full marks *** 14 points.

22. (The full score for this short question is 7)

As shown in the figure, in △ABC, AB=AC is divided into equal parts ∠ABC and DE∨BC.

Proof: de = EC.

23. (The full score for this short question is 7)

As shown in the figure, AB and CD intersect at E, AE=EB, CE=ED, D is the midpoint of straight line FB, and CF and AB intersect at G point. If CF= 15cm, find the length of GF.

Five, this big topic ***2 small questions, full marks *** 16.

24. (The full score for this short question is 8)

As shown in the figure, the parabola y=x2+bx+c intersects with the negative half axis of the X axis at points A and B, intersects with the positive half axis of the Y axis at point C, and intersects with the hyperbola y= is (1, m), and OA=OC. Find the analytical formula of parabola.25. (Full score for this little question)

In May this year, an engineering team (Group A and Group B) contracted the subgrade reconstruction project of the middle section of Renmin Road, which was originally scheduled to be completed in a few days.

(1) It is known that the time required for group A to complete this project alone is twice as long as the specified time by 4 days, while the time required for group B to complete this project alone is twice as short as the specified time 16 days. If the merger of Party A and Party B is completed within 24 days, can it be completed within the specified time?

(2) In actual work, after Party A and Party B completed the project, the construction team contracted the reconstruction project of the eastern section, and one group needed to be transferred. Considering the timely completion of intermediate tasks, which group do you think is the best? Please explain the reason.

Six, this big question * * 1 small question, full score ***9 points.

26. (The full score for this short question is 9)

Read the following materials and solve the following problems.

In the acute angle △ABC, the opposite sides of ∠A, ∠B and ∠C are A, B and C respectively. If A is AD⊥BC in D (as shown in the figure), then sinB=, sinC=, that is, AD=csinB and AD=bsinC, so csinB=bsinC.

So ............ (*)

That is to say, in a triangle, the ratio of the sine value of each side to its diagonal is equal.

(1) If the three elements A, B and ∠A in an acute triangle are known, the above conclusion (*) and related theorems can be applied.

To find the other three unknown elements c, ∠B and ∠C, please fill in the blanks according to the following steps to complete the solution process:

Step 1: From conditions A, B, ∠ A ∠A∠B；;

Step 2: Conditions ∠A, ∠ B. ∠b .∠C；;

Step 3: According to the conditions.

(2) A cargo ship measured that lighthouse A was 30 northwest of the cargo ship at C, then the cargo ship sailed 45 northeast at a speed of 28.4 knots, and arrived at B half an hour later. At this time, lighthouse A was 70 northwest of the cargo ship (as shown in figure 1 1). Find the distance AB (the result is accurate to 0. 1. Reference data: SIN40 = 0.643, SIN65 = 0.906, sin70 =0.940, SIN75 = 0.966).

Seven, this big topic * *1small topic, full score *** 1 0.

27. (The full score of this short question is 1O)

As shown in the figure, the coordinates of a and b are (x 1, 0) and (x2, 0) respectively, where x 1 and x2 are two of the equations x2+2x+m-3=O about x, and x 1

(1) Find the range of m;

(2) Set point C, ∠ ACB = 90, ∠ CAB = 30 on the positive semi-axis of Y axis to find the value of m;

(3) Under the above conditions, if point D is in the second quadrant △ DAB △ CBA, find the resolution function of straight line AD:

Eight, this big topic * *1small topic, full mark *** 1 2.

28. (The full score of this short question is 12)

As shown in the figure (1), AB is the diameter ⊙O, ray AT⊥AB, point P is the moving point on ray AT (P does not coincide with A), PC and ⊙O are tangent to C, and C is CE⊥AB in E, connecting BC and BC intersection point A to T point D, and connecting PB intersection point D.

(1) Please write down the relationship between PA and PD and explain the reasons;

(2) Please find out which triangles are divided into two parts by PB and prove them;

(3) Let the radius of the circle passing through points A, C and D be r, and when CF=R, find the number of times ∠APC, and make a little p in Figure (2) (a ruler is required to draw, but traces of drawing should be left). The answer of mathematics examination questions in Yulin city in 2005.

Reference answers and grading standards of mathematics test papers

Fill in the blanks (2 points for each small question, 20 points for * * *)

The title is 1 2 3 4 5 6 7 8 9 lO.

The answer -4 A x≤3 circumscribes 70 Y2-Y-2 = O x = 2, Y = 3 or x=3, y = 212.66021o.

Second, multiple-choice questions (3 points for each small question, ***24 points)

The title is112131415161718.

Answer D A B C D B C A

Three. 19.5

20. It can be seen that only the sign of the score itself is different between A and B (4 points).

Therefore, a and b are opposite numbers. (5 points)

2l。 A: 85,53.2.

B: 85,70.4.

As can be seen from the above data, the math score of student B is not stable enough and fluctuates greatly. I hope that student B can fill in the gaps in learning and strengthen the ability training.

22. Proof: DB/AB=EC/AC( 1) because of DE∑BC.

AB=AC, so DB=EC(3 points)

Because de ∥BC, so ∠ DE =∠EBC(4 points)

And ∠DBE=∠EBC, so ∠ Deb = ∠ DBE. (5 points)

So db = de. (6 points)

So DE=EC (7 points)

23.GF = 10 (cm). (7 points)

24. solution: substitute x= 1, y=m, y=6/x to get m = 6. ( 1)

Substitute x= 1 and y=6 into y=x2+bx+c to get B+C = 5. ① (2 points).

Let x=O and y=c, then the coordinate of point c is (0, c). (3 points)

And OA=OC, so the coordinate of point A is (-c, o). (4 points)

So (-c)2+b(-c)+c=O, and c >；; 0，c-b =- 1。 ② (5 points)

Solve the equation formed by ① and ② and get b=3c=2.

So y = x2+3x+2. (8 points)

25. Solution: (1) If the specified time is X days, then

X 1=28, x2 = 2. (3 points)

The test shows that x 1=28 and x2=2 are the roots of the original equation.

But x2=2 doesn't meet the problem, so I'll take x = 28.

By the age of 24

(2) It will take y days to set up a team A and B to complete 5/6 of this project.

rule

The solution gives y=20 (days). (5 points)

Time required for Party A to complete the remaining work alone: 10 (days).

Because 20+l0 = 30 >: 28,

Therefore, Party A cannot complete the remaining projects within the specified time; (6 points)

Time required for Party B to complete the remaining work alone: 20/3 (days).

Because 20+20/3 = 26

Therefore, Party B can complete the remaining projects within the specified time. (7 points)

So I think it's best to transfer to group A (8 points)

26. Solution: (1), ∠A+∠B+∠C = 180, A, ∠A, ∠C or B, ∠B, ∠C,

or

(2) According to the meaning of the question ∠ ABC = 65,

∠ A = 40。 (5 points)

BC = 14.2。 (6 points)

AB≈2 1.3。

A: The distance between the freighter and Lighthouse A is about 2 1.3 nautical miles. (9 points)

27. Solution: (1) Judging from the meaning of the question, yes.

22-4(m-3)= 16-m & gt； 0①

x 1x 2 = m-3 & lt； O. ②

① Get an M.

M

So the range of m is

(2) From the meaning of the question, we can get ∠ OCB = ∠ CAB = 30.

So BC=2BO, AB = 2bc = 4bo.

So A0=3BO(4 points)

Thereby obtaining x 1 =-3x2. ③.

And because x 1+x2 =-2.

Combining the solutions of ③ and ④, x 1=-3, X2 = 1. (5 points)

Substitute x 1 x2 = m-3 to get m = O. (6 points)

(3) If the axis of DF⊥ intersects with d, it is f 。

According to (2), the coordinates of a and b are A(-3, o) and B( 1, o) respectively.

So BC=2, AB=4, OC=

Because △ dab △ CBA,

So DF=CO=, AF=B0= 1, of = A0-af = 2.

So the coordinate of point D is (-2,).

The resolution function of the straight line AD is y=x=3.

28. Solution: (1) Link AC.

Because AT⊥AB, AB is a diameter ⊙O,

So a t is the tangent of ⊙ O.

PC is the tangent of ⊙O,

So pa = PC.

So ∠ PAC = ∠ PCA.

Because AB is the diameter of ⊙O,

So ∠ ACB = 90

So ∠ PAC+∠ ADC = 90, ∠ PCA+∠ PCD = 90.

So ∠ADC=∠PCD. ..

So PD = PC = pa.

(2) According to (1), PD=PA, with the same height, it can be seen that △ABD is divided into two triangles with equal areas by PB.

Because AT⊥AB, CE⊥AB,

So in ∨ce.

So CF/PD=BF/BP, EF/PA = BF/BPF.

So cf/PD = ef/pa.

So cf = ef. (6 points)

It can be seen that △CEB is also divided into two triangles with equal areas by PB. (7 points)

(3) According to (1), PA=PCPD,

So PA is the radius of the circumscribed circle of △ACD, that is, pa = r.

According to (2), CF=EF and CF= 1/4 R,

So ef = 1/4 pa.

So ef/pa = 1/4.

Because EF∑AT, BE/AB=EF/PA= 1/4.

So CE== BE

In Rt△ACE,

Because tan CAE =/3.

So CAE = 30.

So PAC = 90-CAE = 60.

And PA=PC, so △PAC is an equilateral triangle.

So APC = 60.

The drawing of point p is shown in the figure.